The de Jong fundamental group of a non-trivial abelian variety is non-abelian

We show that the de Jong fundamental group of any non-trivial abelian variety over a complete algebraically closed extension $C\mathrm{/}{\mathbb{Q}}_p$ is non-abelian. Generalizing an argument for ${\mathbb{P}}_C^1$, we also show that the de Jong fundamental group of any connected rigid analytic variety over $C$ admitting a non-constant map to ${\mathbb{P}}_C^n$ (e.g., an abelian variety) depends on $C$ and is big.